Older Brother

Your older brother is an amateur mathematician with lots of
experience. However, his memory is very bad. He recently got
interested in linear algebra over finite fields, but he does
not remember exactly which finite fields exist. For you, this
is an easy question: a finite field of order $q$ exists if and only if $q$ is a prime power, that is,
$q = p^ k$ holds for some
prime number $p$ and some
integer $k \geq 1$.
Furthermore, in that case the field is unique (up to
isomorphism).